Obsolescence of condition 1 in the inverse spectral theorem

The paper concerns a first-order system on the half line with spectral data satisfying the conditions of Theorem 4.1; condition 1 is one of the hypotheses in that theorem.

Obsolescence conjecture. Condition 1 in Theorem 4.1 is obsolete in general, meaning that the theorem should remain valid without this condition.

This is a claim about removing a hypothesis from the paper's main inverse spectral result. The supplied text gives no proof or resolution, so whether condition 1 can be omitted in general remains open.